Final answer:
A function is discontinuous at x values where it is undefined, goes to infinity (creating an asymptote), or experiences a jump. An example is the function y = 1/x, which is discontinuous at x = 0 due to a vertical asymptote.
Step-by-step explanation:
The question asks at which x value(s) a function is discontinuous. Discontinuities in a function can occur for several reasons, such as when there is a gap in the function (where the function is undefined or not continuous), at points where the function goes to infinity (creates an asymptote), or where there is a jump discontinuity (the value of the function suddenly changes at a point).
For example, in the function y = 1/x, the function is discontinuous at x = 0 because as x approaches zero, y approaches infinity, which is an example of a vertical asymptote. Similar characteristics of discontinuity can be observed in other functions where they approach infinity or have undefined points within a given domain.
For a function to be continuous, the function itself must be continuous, and its first derivative must also be continuous, unless there is a point where the potential function V(x) = ∞. In the context of continuous probability density functions, discontinuity would mean that the probability at a given point does not exist, as the function needs to be integrable over a range to yield a meaningful probability